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Maximum modulus, characteristic, and area on the sphere. (English) Zbl 0703.30025
Let f be an entire function of infinite order. T(r) its Ahlfors-Shimizu characteristic. If \(\gamma\) (r) is a differentiable, increasing function such that T(r)\(\leq \gamma (r)\) \((r>r_ 0)\), then
(a) \(\liminf_{r\to \infty}(\log M(r,f)/r\gamma '(r))\leq \pi;\)
(b) If log \(\gamma\) (r) is a convex function of log r, \(\limsup_{r\to \infty}(\log M(r,f)/r\gamma '(r))\leq \pi.\)
The choice \(\gamma (r)=T(r)\) together with known results [N. V. Govorov, Funkts. Anal. Prilozh. 3, No.2, 41-45 (1969; Zbl 0197.053)] yields:
For every entire function of order \(\geq 1/2\) \[ \liminf_{r\to \infty}\log M(r,f)/A(r)\leq \pi. \] Here A(r) is the area of the image of \(\{| z| \leq r\}\) under the map f: \({\mathbb{C}}\to Riemann\) sphere.
The paper also identifies increasing functions \(\psi\) such that \[ \liminf_{r\to \infty}\log M(r,f)/\Psi (T(r,f))\leq \pi. \] The proofs are based on a clever application of real variable results to the Baernstein \(T^*\) function.
Reviewer: W.H.J.Fuchs

MSC:
30D20 Entire functions of one complex variable, general theory
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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